A fixed-point proximity algorithm for recovering low-rank components from incomplete observation data with application to motion capture data refinement

Abstract

Low-rank matrix recovery is an ill-posed problem increasingly involved and treated vitally in various fields such as statistics, bioinformatics, machine learning and computer vision. Robust Principle Component Analysis (RPCA) is recently presented as a 2-terms convex optimization model to solve this problem. In this paper a new 3-terms convex model arising from RPCA is proposed to recover the low-rank components from polluted or incomplete observation data. This new model possesses three regularization terms to reduce the ill-posedness of the recovery problem. Essential difficulty in algorithm derivation is how to deal with the non-smooth terms. The ALM method is introduced to solve the original 2-terms RPCA model with convergence guarantee. However, for solving the proposed 3-terms model, its convergence is no longer guaranteed. As a different approach based on fixed point theory, we introduce the proximity operator to handle nonsmoothness, and consequently a new algorithm derived from Fixed-Point Proximity Algorithm (FPPA) is proposed with convergence analysis. Numerical experiments on the problems of RPCA and Motion Capture Data Refinement (MCDR) demonstrate the outstripping effectiveness and efficiency of the proposed algorithm.